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McKay-Thompson series of class 42B for Monster.
1

%I #25 Jun 28 2018 05:22:51

%S 1,0,1,0,0,-2,4,-2,0,0,1,-4,5,-4,4,-2,4,-10,12,-12,8,-4,9,-16,21,-22,

%T 13,-8,17,-32,44,-40,25,-24,38,-56,70,-66,50,-44,57,-92,124,-116,89,

%U -80,106,-164,203,-188,151,-132,173,-264,316,-298,250,-230,300

%N McKay-Thompson series of class 42B for Monster.

%H G. C. Greubel, <a href="/A058672/b058672.txt">Table of n, a(n) for n = -1..2500</a>

%H D. Ford, J. McKay and S. P. Norton, <a href="http://dx.doi.org/10.1080/00927879408825127">More on replicable functions</a>, Commun. Algebra 22, No. 13, 5175-5193 (1994).

%H David A. Madore, <a href="http://mathforum.org/kb/thread.jspa?forumID=253&amp;threadID=1602206&amp;messageID=5836094">Coefficients of Moonshine (McKay-Thompson) series</a>, The Math Forum

%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>

%F Expansion of 2 + ((eta(q)*eta(q^6)*eta(q^14)*eta(q^21))/(eta(q^2)* eta(q^3)*eta(q^7)*eta(q^42)))^2 in powers of q. - _G. C. Greubel_, Jun 24 2018

%F a(n) ~ -(-1)^n * exp(2*Pi*sqrt(n/21)) / (2 * 21^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Jun 28 2018

%e T42B = 1/q + q - 2*q^4 + 4*q^5 - 2*q^6 + q^9 - 4*q^10 + 5*q^11 - 4*q^12 + ...

%t eta[q_] := q^(1/24) * QPochhammer[q]; A := ((eta[q] * eta[q^6] * eta[q^14] * eta[q^21])/(eta[q^2] * eta[q^3] * eta[q^7] * eta[q^42]))^2; a := CoefficientList[Series[2 + A, {q, 0, 60}], q]; Table[a[[n]], {n, 50}] (* _G. C. Greubel_, Jun 24 2018 *)

%t nmax = 60; CoefficientList[Series[2*x + Product[((1 + x^(3*k))*(1 + x^(7*k))/((1 + x^k)*(1 + x^(21*k))))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jun 28 2018 *)

%o (PARI) q='q+O('q^70); F = 2 + ((eta(q)*eta(q^6)*eta(q^14)*eta(q^21))/( eta(q^2)*eta(q^3)*eta(q^7)*eta(q^42)))^2/q; Vec(F) \\ _G. C. Greubel_, Jun 24 2018

%Y Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.

%K sign

%O -1,6

%A _N. J. A. Sloane_, Nov 27 2000

%E More terms from _Michel Marcus_, Feb 24 2014